L(s) = 1 | + (−0.707 + 1.22i)2-s + (−0.999 − 1.73i)4-s + (−2.15 + 4.51i)5-s + (5.04 + 2.91i)7-s + 2.82·8-s + (−4 − 5.83i)10-s + (−14.2 − 8.24i)11-s + (−7.14 + 4.12i)14-s + (−2.00 + 3.46i)16-s − 11.3·17-s + 12·19-s + (9.96 − 0.775i)20-s + (20.1 − 11.6i)22-s + (−12.0 − 20.8i)23-s + (−15.6 − 19.4i)25-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.431 + 0.902i)5-s + (0.721 + 0.416i)7-s + 0.353·8-s + (−0.400 − 0.583i)10-s + (−1.29 − 0.749i)11-s + (−0.510 + 0.294i)14-s + (−0.125 + 0.216i)16-s − 0.665·17-s + 0.631·19-s + (0.498 − 0.0387i)20-s + (0.918 − 0.530i)22-s + (−0.522 − 0.905i)23-s + (−0.627 − 0.778i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.413 + 0.910i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.413 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5065692414\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5065692414\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 1.22i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.15 - 4.51i)T \) |
good | 7 | \( 1 + (-5.04 - 2.91i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (14.2 + 8.24i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 11.3T + 289T^{2} \) |
| 19 | \( 1 - 12T + 361T^{2} \) |
| 23 | \( 1 + (12.0 + 20.8i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-16 - 27.7i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 23.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (49.9 - 28.8i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (35.3 + 20.4i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-17.6 + 30.6i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 67.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-14.2 + 8.24i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-8 + 13.8i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-5.04 + 2.91i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 116. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (-36 + 62.3i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-21.9 + 37.9i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 65.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (141. + 81.6i)T + (4.70e3 + 8.14e3i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.06372774615341020494590356557, −8.652607676700973553795781441689, −8.246773893891077187760081728809, −7.38177114041431807554144959745, −6.53531107107094703437885391266, −5.55367799495795713612726075484, −4.70342096140153723240161635152, −3.29911784583145984509211723343, −2.17169935596389801538577890400, −0.20487223769720513872163339257,
1.22476300992649258849215577222, 2.39361733205909810586739799038, 3.85133355127421471648040228285, 4.73088964002327242399674216415, 5.43375898823392580501123366094, 7.12326572511735349812000158373, 7.908718948147017619138935871442, 8.389502194978815356081043915521, 9.510308666054898605708889640501, 10.13663544080566960379067902335